1.1 Introduction

Tools to deal with many-body systems at finite temperature were developed long ago. As early as 1932 R. E. Peierls ^[1] established the general framework of thermodynamic perturbation theory for quantum systems. In 1955 T. Matsubara ^[2] worked out the details of thermal perturbation theory for many-body systems (see also Ref. ^[3]). Several textbooks in the field have by now been available for some time. Among these we wish to mention those of Baym and Kadanoff, ^[4] Abrikosov, Gorkov, and Dzyaloshinsky, ^[5] Thouless, ^[6] Fetter and Walecka. ^[7] More recent reviews including a presentation of functional methods ^[8] can be found in the monographs by Negele and Orland ^[9] and Blaizot and Ripka. ^[10] It is also worthwhile mentioning some of the classic articles in the field such as the elegant construction given by M. Gaudin in 1960 ^[11] of Wick's theorem at finite temperature, which exploits the algebra of exponentials of quadratic forms in the field operators. As a second example we mention the exhaustive discussion of the mean-field approximation (including small-amplitude collective motion in hot Fermi systems) given by des Cloizeaux in his 1967 les Houches lectures ^[12] using the framework of the variational principle.

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The motivation for the present chapter arises from the renewed interest in finite temperature many-body methods which has appeared recently...